Logarithms of Negative and Imaginary Numbers

By *Euler's identity*,
, so that

from which it follows that for any , .

Similarly, , so that

and for any imaginary number , , where is real.

Finally, from the polar representation for complex numbers,

where and are real. Thus, the log of the magnitude of a complex number behaves like the log of any positive real number, while the log of its phase term extracts its phase (times ).

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